WASHINGTON—President Donald Trump on Wednesday embraced a Senate proposal to cut the number of green cards issued annually by half, as part of his drive to reduce legal as well as illegal immigration into the U.S.

The measure, sponsored by Republican Sens. Tom Cotton of Arkansas and David Perdue of Georgia, would maintain the existing number of employment-based green cards issued each year, but would sharply reduce immigration based on family ties, and it would end a lottery that gives people from underrepresented countries...